Nielsen and Chuang Exercise 2.34

Posted in Nielsen/Chuang by rpmuller on March 9, 2010

Exercise 2.34 Find the square root and the logarithm of the matrix

$\displaystyle \left[\begin{array}{rr} 4 & 3 \\ 3 & 4 \end{array}\right].$

The eigendecomposition of the matrix is:

$\displaystyle \left[\begin{array}{rr} 4 & 3 \\ 3 & 4 \end{array}\right] = \frac{1}{2} \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right] \left[\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right] \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right]$

Thus, the square root and logarithm of the matrix are:

$\displaystyle \sqrt{M} = \frac{1}{2} \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right] \left[\begin{array}{rr} 1 & 0 \\ 0 & \sqrt{7} \end{array}\right] \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right]$

$\displaystyle \log(M) = \frac{1}{2} \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right] \left[\begin{array}{rr} 0 & 0 \\ 0 & \log(7) \end{array}\right] \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right]$

S	M	T	W	T	F	S
	1	2	3	4	5	6
7	8	9	10	11	12	13
14	15	16	17	18	19	20
21	22	23	24	25	26	27
28	29	30	31

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